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Stability properties of steady-states for a network of ferromagnetic nanowires
Stéphane Labbé, Yannick Privat, Emmanuel Trélat
To cite this version:
Stéphane Labbé, Yannick Privat, Emmanuel Trélat. Stability properties of steady-states for a network of ferromagnetic nanowires. Journal of Differential Equations, Elsevier, 2012, 253 (6), pp.1709-1728.
�10.1016/j.jde.2012.06.005�. �hal-00492758v2�
Stability properties of steady-states for a network of ferromagnetic nanowires
St´ephane Labb´e∗ Yannick Privat† Emmanuel Tr´elat‡
Abstract
We investigate the problem of describing the possible stationary configurations of the magnetic moment in a network of ferromagnetic nanowires with length L con- nected by semiconductor devices, or equivalently, of its possibleL-periodic stationary configurations in an infinite nanowire. The dynamical model that we use is based on the one-dimensional Landau-Lifshitz equation of micromagnetism. We compute all L-periodic steady-states of that system, define an associated energy functional, and these steady-states share a quantification property in the sense that their energy can only take some precise discrete values. Then, based on a precise spectral study of the linearized system, we investigate the stability properties of the steady-states.
Keywords: Landau-Lifshitz equation, steady-states, elliptic functions, spectral theory, stability.
1 Introduction
Ferromagnetic materials are nowadays in the heart of innovating technological applica- tions. A concrete example of current use concerns magnetic storage for hard disks, mag- netic memories MRAMs or mobile phones. In particular, the ferromagnetic nanowires are objects that establish themselves in the domain of nanoelectronics and in the conception of the memories of the future. Indeed, the storage of magnetic bits all along nanowires seems to be a promising option not only in terms of footprint but also in terms of speed access to the informations (see [23, 24]). The conception of three dimensional memories based on the use of spin injection permits to hope access millions times shorter than the one observed nowadays in hard disks. In view of such potential application issues to rapid magnetic recording, it is of interest to be able to describes all possible stationary configu- rations of the magnetic moment and to investigate their natural stability properties; this is also a first step towards potential control issues, where the control may be for instance
∗Univ. Grenoble, Laboratoire Jean Kuntzmann, Tour IRMA, 51 rue des Math´ematiques, BP 53, 38041 Grenoble Cedex 9, France; [email protected]
†ENS Cachan Bretagne, CNRS, Univ. Rennes 1, IRMAR, av. Robert Schuman, F-35170 Bruz, France;
‡Universit´e Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France; [email protected]
an external magnetic field, or an electric current crossing the magnetic domain, in order to act on the configuration of the magnetic moment.
The most common model used to describe the static behavior of ferromagnetic mate- rials was introduced by W.-F. Brown in the 60’s (see [4]). From this point of view, the equilibrium states of the magnetization are seen as the minimizers of a given functional energy, consisting of several components. When we consider a ferromagnetic material oc- cupying a domain Ω⊂R3, characterized by the presence of a spontaneous magnetization m almost everywhere, of norm 1 in Ω, the associated energy E(m) takes the form (see [13])
E(m) =A Z
Ω|∇m|2dx− Z
Ω
Ha·mdx+1 2
Z
R3
|Hd(m)|2dx, (1) and other relevant terms can be added for a more accurate physical model (e.g. anisotropic behavior of the crystal composing the ferromagnetic material) but these terms already explain a wide variety of phenomena. The first term is usually called “exchange term”, and A > 0 is the exchange constant. The second term is the external energy, resulting from the possible presence of an external magnetic field Ha and the last term is the so- called “demagnetizing-field”, which reflects the energy of the stray-field Hd(m) induced by the distribution mand is obtained by solving
div(Hd+m) = 0 inD′(R3),
curl(Hd) = 0 inD′(R3), (2)
where m is extended toR3 by 0 outside Ω, andD′(R3) denotes the space of distributions on R3.
The dynamical aspects of micromagnetism are usually described by theLandau-Lifshitz equation introduced in the 30’s in [21], written as
∂m
∂t =−m∧He(m)−m∧(m∧He(m)), (3) where m(t, x) is the magnetic moment of the ferromagnetic material at time t, and He = 2A∆u+Hd(u) +Ha is called the effective field. The existence of global weak solutions of that equation has been studied in [3, 28]. Results on strong solutions locally in time and initial data have been derived in [9]. For more details about modelization, stability and homogenization properties, we refer the reader to [10, 11, 12, 13, 14, 15, 26, 27, 28]
and references therein. Numerical aspects have been investigated e.g. in [1, 11, 20], and control issues using such models have been addresses in [2, 7, 8] for particular magnetic domains.
Notice that, given a solution mof (3), there holds d
dt(E(m(t,·)) =− Z
ΩkHe(m(t, x))− hHe(m(t, x)), m(t, x)im(t, x)k2dx,
and thus this energy functional is naturally nonincreasing along a solution of (3). Every steady-state of (3) must satisfy m∧He(m) = 0 since both terms appearing in the right- hand side of (3) are orthogonal, and as expected the set of steady-states coincides with extremal points of the energy functional (1).
In this article, we consider a one-dimensional model of a ferromagnetic nanowire, for which Γ convergence arguments permit to derive the one-dimensional version of the Landau-Lifshitz equation
∂u
∂t =−u∧h(u)−u∧(u∧h(u)), (4) (see [26], see also [6] for arguments concerning a finite length nanowire) whereu(t, x)∈R3 denotes the magnetization vector, for every x ∈ R and every time t (recall that it is a unit vector), and where h(u) = ∂∂x2u2 −u2e2 −u3e3 (assuming without loss of generality A = 1/2). Here, (e1, e2, e3) denotes the canonical basis of R3 and the nanowire coincides with the real axis Re1.
Given a positive real numberL, our aim is to obtain a complete description of the L- periodic steady-states of (4) and to investigate their stability properties. The motivation of this question is double. First, the equation above, combined with L-periodic conditions on u and ∂u∂x, is the limit model for a straightline network of ferromagnetic nanowires of length L, connected by semiconductor devices. In that case, the period L is imposed by the physical setting. Second, our study will provide a description of all possible periodic steady-states of an infinite length one-dimensional ferromagnetic nanowire, which can be seen as the limit case of L-periodic steady-states in a finite length nanowire where L is very small compared with the length of the nanowire. Note that the authors of [5] have studied particular steady-states called travelling walls for straight ferromagnetic nanowires of infinite length. In [6], the stability of one particular steady-state is investigated in a finite length nanowire with Neumann boundary conditions.
The article is organized as follows. We compute all possible L-periodic steady-states of (4) in Section 2 and prove that they share an energy quantification property, in the sense that their energy can only take isolated values. The stability properties of these steady-states are investigated in details in Section 3, based on a spectral study of the linearized system. Section 4 is devoted to the proof of our main result on quantification.
2 Computation of all periodic steady states
In what follows, the prime stands for the derivation with respect to the space variable x, and S2 denotes the unit sphere of R3 centered at the origin.
Definition 1. A L-periodic steady-state of (4) is a functionu∈C2(R,S2) such that u∧h(u) = 0 on (0, L),
u(0) =u(L), u′(0) =u′(L). (5) Denoting as previously (e1, e2, e3) the canonical basis of R3, with the agreement that the nanowire coincides with the axisRe1, every steady-state can be written asu=u1e1+
u2e2+u3e3, and (5) yields
u1u′′3−u′′1u3−u1u3= 0 on (0, L), u2u′′3−u3u′′2 = 0 on (0, L), u1u′′2−u′′1u2−u1u2= 0 on (0, L), u21+u22+u23 = 1 on (0, L), u(0) =u(L), u′(0) =u′(L).
(6)
The integration of the second equation of (6) yields the existence of a real numberα such that u2u′3−u′2u3=α on [0, L]. Moreover, sinceu takes its values in S2, we set
u1(x) = cosθα(x),
u2(x) = cosωα(x) sinθα(x), u3(x) = sinωα(x) sinθα(x),
(7)
for every x∈R. Then, we infer from (6) that
2θ′′αsinωα+ωα′′cosωαsin(2θα)−(ω′α2+ 1) sinωαsin(2θα) + 4ω′αθα′ cosωαcos2θα= 0, 2θ′′αcosωα−ωα′′sinωαsin(2θα)−(ω′α2+ 1) cosωαsin(2θα)−4ωα′θα′ sinωαcos2θα= 0, ωα′ sin2θα =α,
θα(0) =θα(L) mod 2π, θα′(0) =θ′α(L), ωα(0) =ωα(L) mod 2π, ω′α(0) =ω′α(L).
(8)
Multiplying the first equation by sinωα, the second one by cosωα and adding these two equalities, it follows that (θα, ωα) is solution of
ωα′ sin2θα =α,
−θ′′α+1
2 ω′α2+ 1
sin(2θα) = 0, θα(0) =θα(L) mod 2π, θα′(0) =θ′α(L), ωα(0) =ωα(L) mod 2π, ω′α(0) =ω′α(L).
(9)
At this step, the parameterαplays a particular role. First of all, observe that, if there exists x0 ∈[0, L] such that sin2θα(x0) = 0, then there must holdα = 0. In that case, ω0 is constant, andθ0 satisfies the pendulum equation
θ′′0− 1
2sin(2θ0) = 0, (10)
with periodic boundary conditions
θ0(0) =θ0(L) mod 2π, θ0′(0) =θ0′(L). (11) The case α6= 0 can only occur provided sin2θα(x) >0, for every x∈[0, L]. In that case, we infer from (9) that θα satisfies the equation
θα′′− 1 2
α2 sin4θα + 1
sin(2θα) = 0, (12)
with periodic boundary conditions
θα(0) =θα(L) mod 2π, θα′(0) =θ′α(L). (13) Remark 1. Note that, for every solution θα of (12), the function
x7→θα′(x)2+ α2
sin2θα(x) + cos2θα(x) is constant, and we define the functional
Eα(θα) =θ′α2+ α2 sin2θα
+ cos2θα. (14)
It is related to the energy defined by (1) in the following way. Let u be a steady-state, associated with (θα, ωα) by the formula (7), where θα and ωα are solutions of (9). Then the energy E(u) defined by (1) is given by
E(u) = 1 2
Z L 0
θ′α(x)2+ α2
sin2θα(x)+ cos2θα(x)
dx= L
2Eα(θα). (15) In Section 4, we prove the following result.
Theorem 1. The set of real numbers α for which there exists a steady-state (θα, ωα) consists of isolated values, and contains in particular α = 0. Furthermore, if α denotes any of these isolated values, there exists a family (En)n∈N∗ such that Eα(θα)∈ {En}n∈N∗. The proof of that result is quite long and technical, and is postponed to Section 4.
Notice that, using Remark 1, the energy of any steady-state uα withα6= 0 is greater than the energy of any steady-state u0 withα= 0, that is,
E(uα)> E(u0).
This property makes steady-states with α= 0 of particular interest, and in the sequel we focus on them. We next provide a precise description of all steady-states with α = 0. In that case, θ0 is solution of the pendulum equation (10), the solutions of which are well known in terms of elliptic functions (see [22]), as recalled next.
First of all, recall that, for every solutionθ0of (10), the functionx7→θ′0(x)2+cos2θ0(x) is constant, and the value of the constant is E0(θ0).
Recall that, given k ∈(0,1), ˜k=√
1−k2 and η ∈ [0,1], the Jacobi elliptic functions cn, sn and dn are defined from their inverse functions with respect to the first variable,
cn−1 : (η, k) 7−→
Z 1
η
q dt
(1−t2)(˜k2+k2t2) sn−1 : (η, k) 7−→
Z η
0
p dt
(1−t2)(1−k2t2) dn−1 : (η, k) 7−→
Z 1
η
p dt
(1−t2)(t2+k2−1) (η >p
1−k2 in that case)
and the complete integral of the first kind is defined by K(k) =
Z π/2 0
p dθ
1−k2sin2θ.
The functions cn and sn are periodic with period 4K(k) while dn is periodic with period 2K(k).
Using these elliptic functions, solutions of (10) can be integrated as follows, depending on the value of the energy E0(θ0).
IfE0(θ0) = 0, then θ0(x) = π2 for everyx∈[0, L].
If 0<E0(θ0)<1, then
θ0′(x) =k cn
x+ sn−1 1
kcosθ(0), k
, k
, (16)
cosθ0(x) =ksn
x+ sn−1 1
kcosθ(0), k
, k
, (17)
for every x ∈ [0, L], with E0(θ0) = k2. The period of θ0 is T = 4K(k) = 4K(p
E0(θ0)).
This case corresponds to the closed curves of Figure 1.
IfE0(θ0) = 1, then
θ0′(x) = 1/cosh x+ argth−1(cosθ(0))
, (18)
cosθ0(x) = tanh x+ argth−1(cosθ(0))
. (19)
This case corresponds to the separatrices (in bold) of the phase portrait drawn on Figure 1.
IfE0(θ0)>1, then
θ′0(x) = 1 kdnx
k + sn−1(cosθ(0), k), k
, (20)
cosθ0(x) = snx
k+ sn−1(cosθ(0), k), k
, (21)
for every x ∈ [0, L], with E0(θ0) = 1/k2. Moreover, θ0(x+T) = θ0(x) + 2π for every x ∈ [0, L] with T = 2kK(k) = 2K(1/p
E0(θ0))/p
E0(θ0). This case corresponds to the curves located above and under the separatrices of Figure 1.
Every steady-state must moreover satisfy the boundary conditions (11), with the period L. These boundary conditions appear as an additional constraint to be satisfied by the solutions above, which turns into a quantification property, as explained in the next result, that makes the conclusion of Theorem 1 more precise.
Theorem 2. (Case α= 0) Set N0 = L
2π
, where the bracket notation stands for the integer part. Then, there exists a family (En)16n6N0 of elements of (0,1) and a countable family (Een)n∈N∗ of elements of (1,+∞) such that, for every steady-state,
• if 06E0(θ0)<1, then E0(θ0)∈ {E1, . . . , EN0};
K1 0 1 2 3 4
K2 K1 1 2
Figure 1: Phase portrait of (10) (in the plane (θ, θ′))
• if E0(θ0)>1, then E0(θ0)∈ {E˜n |n∈N∗}.
Furthermore, there are steady states corresponding to the energy level E0(θ0) = 1.
Remark 2. Note that, if L <2π, there is no solution satisfying E0(θ0)<1.
Remark 3. Using (15), this theorem turns into a quantification property of the physical energies of steady-states.
Proof. To take into account the boundary conditions (11), we have to impose that L is equal to an integer multiple of the period T ofθ0. The expression of T using the elliptic function K has been given previously, depending on the energy E0(θ0). Recall that K is an increasing function from [0,1) into [π/2,+∞). The graph of the periodT as a function of E0(θ0) is given on Figure 2. The conclusion follows easily.
Remark 4. If L tends to +∞ then the steady-state tends to one of the separatrices of Figure (1). Analytically, this means that θ tends to the solution of (18)-(19). This corresponds to the case of an infinite length nanowire and to the steady-state studied in [5, 7].
3 Stability properties of the steady-states with α = 0
In order to investigate the stability properties of the steady-states such that α = 0, we compute the linearized system around a given steady-state and study its spectral properties. In what follows, define the spaces
Hper1 (0, L;R3) = {u∈H1(0, L;R3)|u(0) =u(L)},
Hper2 (0, L;R3) = {u∈H2(0, L;R3)|u(0) =u(L) and u′(0) =u′(L)}.
0 1 2 3 4 5 6 0
2 4 6 8 10 12
Figure 2: Graph of the period T in function of E0(θ0) (case α= 0)
Endowed respectively with the usualH1andH2inner product, these are Hilbertian spaces.
LetM0be a steady-state with α= 0. The results of the previous section show that, in the spherical coordinates (θ, ω) that have been used, the componentωis constant. Clearly, the equation (4) is invariant with respect to rotations around the axis Re1. Then, up to a rotation of angle ω around the axis Re1, we assume that
M0(x) =
cosθ(x) sinθ(x)
0
,
whereθ is solution of (10), (11) as described in Section 2. In Section 3.1, we compute the linearized system around this steady-state. The operator underlying this linearized system is a matrix of one-dimensional operators, one of which, denotedA, plays an important role.
We study in details the spectral properties of A in Section 3.2. Based on this preliminary study, we investigate in Section 3.3 the stability properties of the steady-state M0. Notice that the linearized system is as well invariant with respect to rotations around the axis Re1, and hence these results hold for every L-periodic steady-state. Finally, Section??is devoted to prove that the eigenvalues of the linearized system are simple except for certain discrete values of L.
3.1 Linearization of (4) around a steady-state
Letube a solution of (4). As in [5], we completeM0into the mobile frame (M0(x), M1(x), M2), where M1 and M2 are defined by
M1(x) =
−sinθ(x) cosθ(x)
0
, M2 =
0 0 1
.
Consideringu as a perturbation of the steady-stateM0, since |u(t, x)|= 1 pointwisely, we decompose u:R+×R−→S2⊂R3 in the mobile frame as
u(t, x) = q
1−r21(t, x)−r22(t, x)M0(x) +r1(t, x)M1(x) +r2(t, x)M2. (22) Easy but lengthy computations show that u is solution of (4) if and only if r =
r1 r2
satisfies
∂r
∂t =Lr+R(x, r, rx, rxx), (23) where
R(x, r, rx, rxx) = G(r)rxx+H1(x, r)rx+H2(r)(rx, rx), and
• L =
A+ Id A+E0(θ)Id
−(A+ Id) A+E0(θ)Id
withA =∂xx2 −2 cos2θId defined on the domain D(A) =Hper2 (0, L),
• G(r) is the matrix defined by
G(r) =
r1r2
√1−|r|2
r22
√1−|r|2 +p
1− |r|2−1
−√r12
1−|r|2 −p
1− |r|2+ 1 −√r1r2
1−|r|2
,
• H1(x, r) is the matrix defined by H1(x, r) = 2θ′(x)
p1− |r|2 r2p
1− |r|2−r1r22 −r2(1−r21) r2(1−r22) p
1− |r|2r2+r1r22
,
• H2(r) is the quadratic form onR2 defined by H2(r)(X, X) = (1− |r|2)X⊤X+ (r⊤X)2
(1− |r|)3/2
pp1− |r|2r1+r2
1− |r|2r2−r1
, with the estimates
G(r) =O(|r|2), H1(r) =O(|r|), H2(r) =O(|r|).
It is not difficult to prove that there exists a constant C >0 such that, if |r|2 6 1
2, then, there holds for every x∈R, for every (p, q)∈(R2)2,
|R(x, r, p, q)|6C(|r|2|q|+|r||p|+|r||p|2).
This a priori estimate shows that R(x, r, rx, rxx) is a remainder term in (23).
3.2 Spectral study of the operator A =∂2
xx−2 cos2θ Id
In this section, we derive spectral properties of the operatorAappearing in the expression of the linearized operator L, which will be useful for the stability analysis of Section 3.3.
The domain ofAisHper2 (0, L;R3), but of course it is equivalent to studyAon the domain D(A) =Hper2 (0, L;R) (denoted shortlyHper2 (0, L)).
Every eigenpair (λ, u) of A must satisfy
u′′−2 cos2θ u=λu, u(0) =u(L), u′(0) =u′(L).
This is a particular case of Sturm-Liouville type problems with real coupled self-adjoint boundary conditions (see [17, 18, 19]). The following result provides some spectral prop- erties of A.
Proposition 1. The operator A, defined on D(A) =Hper2 (0, L), is selfadjoint in L2(0, L) and there exists a hilbertian basis (ek)k∈N of L2(0, L), consisting of eigenfunctions of A, associated with real eigenvalues λk that are at most double, with
−∞<· · ·6λk6· · ·6λ1 6λ0, (24) and λk→ −∞ as k→+∞. Moreover,
• the eigenvalue λ0 is simple, and its associated eigenfunction e0 vanishes 0or 1 time on [0, L];
• the eigenfunction ek vanishes k−1 or k or k+ 1 times on [0, L].
Remark 5. A simple computation shows that
Asinθ = −E0(θ) sinθ, Aθ′ = −θ′,
Acosθ = −(1 +E0(θ)) cosθ.
Hence, sinθ, θ′ and cosθ are eigenfunctions of A associated respectively with the eigen- values −E0(θ),−1,−(1 +E0(θ)). We are not able to exhibit nor compute explicitly some other eigenelements of A.
Note that, if the steady-state under consideration satisfies E0(θ) > 1 (that is, the corresponding trajectory on the phase portrait of Figure 1 is outside the separatrices), then the functionθ′ does not vanish, and it follows from Proposition 1 thatλ0 =−1, that is, −1 is the largest eigenvalue of A, and e0 =θ′. Indeed, according to Proposition 1, the function e1 could vanish 0, 1 or 2 times. Nevertheless, this is not the case since the inner product between e0 and e1 must be zero, which indicates that e1 vanishes at least one time.
If the steady-state under consideration satisfies E0(θ) <1 (that is, the corresponding trajectory on the phase portrait of Figure 1 is inside the separatrices), then the function
sinθ does not vanish, and it follows from Proposition 1 thatλ0=−E0(θ), that is,−E0(θ) is the largest eigenvalue of A, and e0 = sinθ.
In the particular case θ = π/2 (corresponding to E0(θ) = 0), one has θ′ = 0 and cosθ = 0 and thus they are not eigenfunctions. In that case, λ0 = 0, and e0 = 1. By the way, all eigenvalues can be easily computed as λk=− 2kπL 2
, and they are all double except for k= 0.
Proof. The proof follows standard arguments. However, we include it from the convenience of the reader. We first prove that the operator Ais diagonalisable. Consider the ordinary differential equation with boundary conditions
−u′′+ (2 cos2θ+ 1)u=f,
u(0) =u(L), u′(0) =u′(L). (25) This problem is equivalent to the problem of determiningu∈Hper2 (0, L) such thatb(u, v) = g(v) for everyv∈Hper1 (0, L), where the bilinear form band the linear form g are defined by
b(u, v) = Z L
0
u′(x)v′(x)dx+ Z L
0
(2 cos2θ(x) + 1)u(x)v(x)dx, g(v) =
Z L
0
f(x)v(x)dx.
Moreover, it is clear that
kuk2H1(0,L) 6 b(u, u),
|b(u, v)| 6 4kukH1(0,L)kvkH1(0,L),
|g(v)| 6 kfkL2(0,L)kvkH1(0,L),
for allu, v ∈Hper1 (0, L). This implies thatbis continuous and coercive, andgis continuous.
Lax-Milgram’s Theorem then implies the existence of a unique weak solution inHper1 (0, L), and it is easy to prove that this solution is strong and belongs to Hper2 (0, L), using a standard bootstrap argument. It is then possible to define the linear operator
F : L2(0, L) −→ L2(0, L) f 7−→ u
where u is the unique solution of (25). The operator F is compact. Indeed, let u=F f, forf ∈L2(0, L). Then,
kuk2H1(0,L)6b(u, u) 6kfkL2(0,L)kukH1(0,L),
and hence kukH1(0,L) =kF fkH1(0,L) 6kfkL2(0,L). Since the imbedding of H1(0, L) into L2(0, L) is compact, it follows that the operator F is compact. For f1, f2 ∈ L2(0, L), denoting u1 =F f1 and u2=F f2, one has
hF f1, f2iL2(0,L)=hu1, f2iL2(0,L) =b(u1, u2) =hf1, F f2iL2(0,L),
and hence, sinceF is bounded onL2(0, L),F is selfadjoint. SinceF is compact and selfad- joint, it follows that the operator Ais diagonalisable with real eigenvalues satisfying (24).
The eigenvalues λk are at most double because the associated eigenfunctions are solutions of a linear ordinary differential equation of order two. There cannot be two successive equalities in (24) because the eigenproblem associated to λn has exactly two linearly in- dependent solutions. The assertions concerning the zero properties of the eigenfunctions follow from [19].
3.3 Stability properties of the steady-states Consider the linear system
∂z
∂t =Lz
z(t,0) =z(t, L), z′(t,0) =z′(t, L),
(26) obtained in Section 3.1 by linearizing the Landau-Lifshitz equation (4) around the steady state M0. As stated in Lemma 1, since (ek)k>0 is a hilbertian basis of L2(0, L) whose elements are eigenfunctions of the operator A, we can write
z(t, x) =
z1(t, x) z2(t, x)
for almost every (t, x)∈R+×(0, L), where zi(t, x) =
+∞
X
k=0
zik(t)ek(x)
for i = 1,2, with zki(t) =hzi(t,·), ekiL2(0,L) for every k ∈N. Then, it is easy to see that (26) is equivalent to the series of 2×2 linear systems
∂zk
∂t =Lkz,
zk(0) =zk(L), zk′(0) =zk′(L), for every k∈N, where
Lk=
λk+ 1 λk+E0(θ)
−(λk+ 1) λk+E0(θ)
.
Recall that a matrix is said Hurwitzian whenever all its eigenvalues have their real part lower than 0. One has the following result.
Lemma 1. For everyk∈N, the matrixLkis Hurwitzian if and only ifλk<min(−1,−E0(θ)).
Proof. Setm= min(−1,−E0(θ)) andM = max(−1,−E0(θ)). The matrixLkis Hurwitzian if and only if its determinant is positive and its trace is negative, that is, if and only if (λk+ 1)(λk+E0(θ))>0 and 2λk+ 1 +E0(θ)<0. The trace condition yieldsλk< m+M2 , and the determinant condition yields λk< m orλk> M. The conclusion follows.
To establish spectral properties of the steady-states, we distinguish between four cases, depending on value of the energy E0(θ) of the steady-state under consideration.
